Design of Differentially Flat Planar Space Robots

Proceedings of the 2003 IEEE/RSJ Intl. Conference on Intelligent Robots and Systems Las Vegas, Nevada · October 2003

Design of Differentially Flat Planar Space Robots: A Step Forward in their Planning and Control Jaume Franch Departament de Matematica Aplicada IV c/ Jordi Girona 1-3, Modul C-3 Campus Nord, UPC 08034-Barcelona Spain

Sunil K. Agrawal, S. Oh, Abbas Fattah Mechanical Systems Laboratory Department of Mechanical Engineering University of Delaware Newark, DE 19716 USA

Abstract— The motion of free-floating space robots is characterized by nonholonomic, i.e., non-integrable rate constraint equations. These constraints originate from principles of conservation of linear and angular momentum. It is well known that these rate constraints can also be written as input-affine drift-less control systems. Trajectory planning of these systems is extremely challenging and computation intensive since the motion must satisfy differential constraints. However, under certain conditions, these drift-less control systems can be shown to be differentially flat. The property of flatness allows a computationally inexpensive way to plan trajectories for the dynamic system between two configurations as well as develop feedback controllers. In this paper, nonholonomic rate constraints for free-floating planar open-chain robots are systematically studied to determine the design conditions under which the system exhibits differential flatness. Under these design conditions, the property of flatness is used for trajectory planning and feedback control under perturbations in the initial state.

[4]. Flat systems include controllable linear systems as well as nonlinear systems linearizable by static and dynamic feedback. For differentially flat systems, a trajectory in the space of flat outputs y(t) and its derivatives is consistent with the dynamics if it satisfies the boundary conditions. The states x(t) and inputs u(t) can be computed from y(t) according to the diffeomorphic relation between the original state variables and flat variables and their derivatives. Differentially flat systems have been studied in the context of trajectory generation with and without auxiliary constraints ([4], [3]). II. PLANAR FREE-FLOATING ROBOT MODELS We consider the modeling of planar free-floating robots in zero gravity space environment. The design consists of a free-floating base with a robot arm mounted on it. The robot arm has an open chain structure. The modeling of the whole system consists of n + 1 rigid links connected by revolute joints where the links are arranged in a serial chain and are numbered from 0 to n as shown in Fig. 1. The first link, i.e., link 0 is the base and we consider the link i in the chain to be connected to link i − 1 at Oi and to i + 1 at Oi+1 . The center of mass of the link i is labeled as Ci and the length of link i is Oi Oi+1 = li . Frame F0 is the inertial frame. Since we assume all external forces and moments on the whole system to be zero, the system center of mass, i.e., C is inertially fixed. Similarly, the net system angular momentum about the inertially fixed point O, i.e., HO = Const [1]. ˆ can be The total angular momentum HO = H k written as n
Hi , (1) H=

I. INTRODUCTION It is well known that input commands to the joints of a free-floating robot cause motion to its base. This happens because of the forces and moments transmitted to the base during manipulator motions. As a result, for the same joint input commands, the motions of a free-floating robot and structurally similar fixed-base robots may be quite different from each other [2]. In the literature, a number of schemes have been proposed to correct for these base motions as the end-effector is commanded to move between desired positions. Motion planning of space robots is in general a tough problem because of their nonholonomic nature. The differential constraints need to be satisfied through the solution of nonlinear programming problems. Hence, most of the schemes for planning tend to be non real-time. A control system x˙ = f (x, u) x ∈ Rn

i=0

u ∈ Rm

where Hi is the contribution to the angular momentum by the ith link. Its expression is

is said to be differentially flat if and only if there exist m functions y = (y1 , . . . , ym ) such that y = y(x, u, u, ˙ . . .) x = x(y, y, ˙ . . .)

0-7803-7860-1/03/$17.00 © 2003 IEEE

Hi = mi (rxi r˙yi − r˙xi ryi ) + Ii

u = u(y, y, ˙ . . .)

i
j=0

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θ˙j ,

(2)

fixed. We get n
i=0 n


mi rxi = M rCx = Const = K1

(10)

mi ryi = M rCy = Const = K2

(11)

i=0

where M =

n  i=0

Fig. 1.

of the position vector of the system center of mass C with respect to O. Upon substitution of rxi and ryi from (8) and (9) into (10) and (11) and simplifying the result, we obtain n n

1 [K1 − mi l0 cos θ0 − mi l1 cos θ01 − . . . x0 = M i=1 i=2

A planar free-floating robot

−mn ln−1 cos θ0(n−1) −

where θ˙j is the j-th joint rate, mi is the mass of link i and Ii is the moment of inertia of link i about its center of mass. The variables rxi and ryi are coordinates of the position vector from point O to the center of mass of link i. Using (2), the total angular momentum of the system can be written for an (n + 1)-link planar free-floating manipulator. The expressions for rxi and ryi are: rxi = ri · ˆi ryi = ri · ˆj,

y0 =

(3) (4)

i=0

(5)

j=0

where rxi and ryi are defined in (8) and (9). For the sake of simplicity we have considered αi = 0 for all i = 0, . . . , n. One can write x0 and y0 in (8) and (9) for the three link system substituting n = 2 into (12) and (13) in the following way: 1 [K1 − (m1 + m2 )l0 cos θ0 x0 = M 2
− m2 l1 cos(θ0 + θ1 ) − mj dj cos(θ0j )](15) j=0

1 [K2 − (m1 + m2 )l0 sin θ0 y0 = M 2
− m2 l1 sin(θ0 + θ1 ) − mj dj sin(θ0j )],(16)

l0 cos(θ0j ) + di cos(αi + θ0i ), (8)

j=0 i−1


mj dj sin(αj + θ0j )] (13)

III. THREE LINK PLANAR ROBOT The angular momentum equation for a three-link free-floating robot is given inserting n = 2 into (2) as   2 i

mi (rxi r˙yi − r˙xi ryi ) + Ii H= θ˙j  = 0 (14)

where θ0i denotes θ0 + . . . + θi for all i ≥ 1, while θ00 = θ0 . From the expression of ri and using (3) and (4), we get

ryi = y0 +

n
j=0

Oi−1 Oi = li−1 (cos θ0(i−1)ˆi + sin θ0(i−1)ˆj) (6) Oi Ci = di (cos (αi + θ0i )ˆi + sin (αi + θ0i )ˆj) (7)

rxi = x0 +

mj dj cos(αj + θ0j )] (12)

n n

1 [K2 − mi l0 sin θ0 − mi l1 sin θ01 − . . . M i=1 i=2

−mn ln−1 sin θ0(n−1) −

Here, Oi−1 Oi and Oi Ci are

i−1


n
j=0

where ri is ri = OO0 + O0 O1 + . . . + Oi−1 Oi + Oi Ci

mi and rCx and rCy are the coordinates

j=0

l0 sin(θ0j ) + di sin(αi + θ0i ), (9)

where M = m0 + m1 + m2 . On substituting (8), (9), (15), (16) into the angular momentum (14) and collecting the terms, we get (17) H = Ax˙ 0 + B y˙ 0 + C θ˙0 + Dθ˙1 + E θ˙2 = 0,

j=0

where αi is the angle between Oi Oi+1 and Oi Ci and di = Oi Ci shown in Fig. 1. Moreover, x0 and y0 are the coordinates of position vector of point O0 with respect to O. Their expressions can be derived using the fact that the system center of mass is inertially

where the coefficients A, B, C, D, E are given in Appendix. After several computations and simplifications

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using trigonometric identities, the angular momentum equation can be written as

‘one prolongation’ of one of the inputs. The prolonged system has the form θ˙0 = f1 (θ0 , θ1 , θ2 )θ3 + f2 (θ0 , θ1 , θ2 )v2 , θ˙1 = θ3 , (25) θ˙2 = v2 , θ˙3 = v1 ,

[a0 + a1 cos θ1 + a2 cos θ2 + a3 cos θ12 + a4 (k1 cos(θ012 ) + k2 sin(θ012 ))]θ˙0 = [b0 + b1 cos θ1 + b2 cos θ2 + b3 cos θ12 ]θ˙1 + [c0 + c2 cos θ2 + c3 cos θ12 ]θ˙2 , (18)

where v2 = u2 . This new system should be static feedback linearizable. Let us recall, the system m
x˙ = f (x) + gi (x)ui

where the coefficients a0 ,...,a4 , b0 ,...,b3 , c0 ,...,c3 are given in Appendix. A. Drift-less Control Form If we choose joint angles θ1 and θ2 to be actively driven by motors, while θ0 satisfies the constraint, (18) can be also written as θ˙0 = f1 (θ0 , θ1 , θ2 )u1 + f2 (θ0 , θ1 , θ2 )u2 , (19) θ˙1 = u1 , θ˙2 = u2 ,

i=1

is static feedback linearizable if and only if the following distributions are constant rank and involutive [5]: D0 Di

= =

< g1 , . . . , gm > Di−1 + < {[f, g] ∀g ∈ Di−1 } > i ≤ n − 1

and Dn−1 = Rn . The drift and the input vector fields for the system (25) are       0 f2 (θ) f1 (θ)θ3    θ3     g1 =  0  g2 =  0  f =  0   1  0   0 0 1

where b0 + b1 cos θ1 + b2 cos θ2 + b3 cos θ12 Den (20) c0 + c2 cos θ2 + c3 cos θ12 (21) f2 (θ0 , θ1 , θ2 ) = Den where Den = a0 + a1 cos θ1 + a2 cos θ2 + a3 cos θ12 + a4 (k1 cos(θ012 ) + k2 sin(θ012 )).

f1 (θ0 , θ1 , θ2 ) =

IV. DIFFERENTIAL FLATNESS A driftless system with n states and n − 1 inputs is flat if and only if it is controllable [6]. For the system given by (19), the controllability condition formally is to check if the dimension of         f1 (θ) f2 (θ) f1 (θ) f3 (θ) <  1  ,  0  ,  1  ,  0  > 0 1 0 1 (22)      . . is equal to 3, where  .  ,  .  denotes . . Lie of two vector fields. Lie bracket   bracket    f1 (θ) f2 (θ)  1  ,  0  is given by 0 1  ∂f2 (θ) ∂f1 (θ) ∂f2 (θ) ∂f1 (θ)  ∂θ1 − ∂θ2 + f1 (θ) ∂θ0 − f2 (θ) ∂θ0   0 0

Since Lie bracket [g1 , g2 ] vanishes, D0 is trivially involutive. It is a straightforward computation to check   f1 (θ)  1   [f, g1 ] = −   0  0 and [f, g2 ] =  2 (θ) 1 (θ) θ3 (f1 (θ) ∂f∂θ − f2 (θ) ∂f∂θ + 0 0  0 −  0

∂f2 (θ) ∂θ1

−

∂f1 (θ) ∂θ2 )

   

0 Therefore, D1 = R4 if one assumes the controllability of the system. Hence, D1 is also involutive and satisfies the rank condition. For this prolonged system, Kronecker indexes are k1 = #{dim Di − dim Di−1 ≥ 1, i ≥ 0} = 2 k2 = #{dim Di − dim Di−1 ≥ 2, i ≥ 0} = 2 (26) Therefore, the flat outputs y1 , y2 are two differentially independent functions such that

(23) Therefore, system (19) is flat if and only if

∇y1 ⊥ Dk1 −2 = D0 ,

∂f2 (θ) ∂f1 (θ) ∂f2 (θ) ∂f1 (θ) − + f1 (θ) − f2 (θ) = 0 ∂θ1 ∂θ2 ∂θ0 ∂θ0 (24) Moreover, assuming that the system is controllable, a way to obtain the flat outputs is to consider a

∇y2 ⊥ Dk2 −2

(27)

If we denote ∇yi = (ai1 , ai2 , ai3 , ai4 ), the orthogonality condition implies ai4 = 0

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ai1 f2 (θ) + ai3 = 0

(28)

Moreover, in order to ∇y1 and ∇y2 be an exact one form, the following equations must be satisfied: 0=

∂ai4 ∂θ0 ∂ai3 ∂θ0

= =

∂ai1 ∂θ3 ∂ai1 ∂θ2

0=

∂ai4 ∂θ1 ∂ai3 ∂θ1

= =

∂ai2 ∂θ3 ∂ai2 ∂θ2

0=

∂ai4 ∂θ2 ∂ai2 ∂θ0

= =

The functions f1 (θ) and f2 (θ) of equation (25) are b0 + b1 cos θ1 c0 f1 (θ) = f2 (θ) = a0 + a1 cos θ1 a0 + a1 cos θ1 The controllability condition reduces to check  b0 +b1 cos θ1    c0

∂ai3 ∂θ3 ∂ai1 ∂θ1

(29)



Therefore, we can choose ∇y1 = (0, 1, 0, 0)

∇y2 = (a1 , a2 , −a1 f2 (θ), 0)

∂a2 ∂(−a1 f2 (θ)) = ∂θ2 ∂θ1 (30) A solution of this partial differential equations cannot be found in general. It could be easier to solve if the function f2 (θ) does not depend explicitly on some of the variables. In the following section we will consider an specific design of the robot that allows us to compute these flat outputs.

a0 +a1 cos θ1

0 1

 = 0

m1 d1 + m2 l1 = 0.

In order to find the flat outputs, we use the prolongation procedure explained in Section IV. Therefore, we need two functions y1 , y2 such that they are differentially independent and ∇y1 , ∇y2 are orthogonal to     c0 0 a0 +a1 cos θ1  0    0   > D0 =<   0 ,  1 1 0 A set of flat outputs for this system is y1

(34) c0 θ2 (35) y 2 = θ0 − a0 + a1 cos θ1 Let us compute the change of variables and the feedback law needed to linearize the system. First, y˙ 1 = u1 . On differentiating (35) and simplifying, we get

V. DESIGN FOR FLATNESS The objective in this section is to simplify the angular momentum (18) through proper design of the robot such that the PDEs given in (30) can be solved for the flat outputs y1 and y2 . In our design philosophy, we allow the center of mass of the links to be altered by redistribution of the mass. We use a counterweight on the distal link so that its center of mass is on the preceeding joint. Let us consider the case when the center of mass of the third link is on joint two, i.e., the parameter d2 vanishes. Therefore, the angular momentum equation is

y˙ 2

=

= θ1

b0 +b1 cos θ1 a0 +a1 cos θ1 u1

−

c0 a1 sin θ1 u1 (a0 +a1 cos θ1 )2 θ2

.

Therefore, θ2 = −(a0 + a1 cos y1 )2 y˙ 2 + (a0 + a1 cos y1 )(b0 + b1 cos y1 )y˙1 c0 a1 sin y1 y˙ 1 (36) Finally, θ0 is obtained by substituting θ2 in (35):

(31)

−(a0 + a1 cos y1 )y˙ 2 + (b0 + b1 cos y1 )y˙ 1 a1 sin y1 y˙ 1 (37) Hence, the change of variables in the prolonged system is given by (θ0 , θ1 , θ2 , u1 )
=   y˙ 2 +(b0 +b1 cos y1 )y˙ 1 y2 + −(a0 +a1 cos ay11 )sin y1 y˙ 1   y1    −(a0 +a1 cos y1 )2 y˙ 2 +(a0 +a1 cos y1 )(b0 +b1 cos y1 )y˙ 1    c0 a1 sin y1 y˙ 1 y˙ 1 (38) The feedback law is given as (u˙ 1 , u2 )
=

  1 0 y¨1 = (a0 +a1 cos y1 )2 y˙ 2 (a0 +a1 cos y1 )2 − c0 a1 sin y1 y˙ 1 y¨2 c0 a1 sin y1 y˙ 12   γ1 + (39) γ2 θ0 = y 2 +

where m1 m2 (l1 − d1 )2 + m0 (l0 − d0 )2 (m1 + m2 ) + (I0 + I1 + I2 )M + m0 (m2 l12 + m1 d21 ) a1 = 2m0 (l0 − d0 )(m2 l1 + m1 d1 ) b0 = −(m1 m2 (l1 − d1 )2 + (I1 + I2 )M + m0 (m2 l12 + m1 d21 ) b1 = −(m0 (l0 − d0 )(m2 l1 + m1 d1 ) − m0 m2 d0 l1 ) c0 = −I2 M. (32) This equation can be rewritten as a control system if one considers θ1 and θ2 to be actively driven by motors: c0 1 cos θ1 θ˙0 = ab00 +b +a1 cos θ1 u1 + a0 +a1 cos θ1 u2 (33) θ˙1 = u1 θ˙2 = u2 a0

,

I2 = 0 m0 = 0 l0 = d0

∂a1 ∂a2 = ∂θ1 ∂θ0

[a0 + a1 cos θ1 ] θ˙0 = [b0 + b1 cos θ1 ] θ˙1 + c0 θ˙2 ,

1 0

This reduces to c0 a1 sin θ1 = 0. In other words, the parameters of the system must satisfy

such that a1 and a2 are not functions of θ3 . The main difficulty that one can come across is the integrability of ∇y2 . The integrability conditions for ∇y2 reduce to: ∂a1 ∂(−a1 f2 (θ)) = ∂θ2 ∂θ0

a0 +a1 cos θ1

=

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Fig. 2. The computed trajectories of the two flat outputs y1d , y2d .

Fig. 3.

in Fig. 2. On transforming these trajectories of the flat outputs, the state trajectories are shown in Fig. 3. The geometrical and mechanical properties of the robot in MKS units are: mi = 1, li = 0.2, Ii = 0.0533, dj = lj /2, i = 0, 1, 2, j = 0, 1.

where γ1 γ2

= = − −

The computed trajectories of θ0d , θ1d , θ2d .

0 2(a0 +a1 cos y1 )y˙ 2 −(b0 +b1 cos y1 )y˙ 1 c0 (a0 +a1 cos y1 )b1 y˙ 1 y1 )2 cos y1 y˙ 2 + (a0 +a1ccos c0 a1 0 a1 sin y1 (a0 +a1 cos y1 )(b0 +b1 cos y1 ) cos y1 y˙ 1 c0 a1 sin2 y1

B. Feedback Control As shown in (39), the inputs are related to the flat variables y1 , y2 , which have a relative degree one and two respectively as long as sin(y1 )y˙ 1 = 0. With a feedback law of the form u˙ 1 = v1 (40) u2 = γ2 + γ3 y¨1 + γ4 v2

VI. SIMULATION RESULTS A. Trajectory Generation The trajectory planning problem can be posed as finding a smooth trajectory of the variables θ0 (t), θ1 (t), θ2 (t) between two given end-points over a time t0 to tf . The conditions of θ0 (t0 ), θ1 (t0 ), θ2 (t0 ) can be used in (34)-(35) to compute y1 (t0 ), y2 (t0 ), and a third relationship between their firstorder derivatives at t0 . Similarly, the conditions of θ0 (tf ), θ1 (tf ), θ2 (tf ) can be used in (34)-(35) to compute y1 (tf ), y2 (tf ), and a relationship between the first derivatives of these variables. These boundary conditions can be used to select a smooth trajectory for y(t) between t0 and tf . Now, (38) can be used to compute θ0 (t), θ1 (t), θ2 (t) consistent with the dynamic equations (33). Note that the transformation is nonsingular as long as sin(y1 ) = 0 and y˙ 1 = 0. In the original coordinates, these translate to sin(θ1 ) = 0 and θ˙1 = 0. This must be ensured through the choice of a proper trajectory. A motion plan is developed using the following boundary conditions for the trajectory over 2 seconds of time: θ0 (0) = 1.8, θ1 (0) = 1, θ2 (0) = 0; θ0 (2) = 1.2, θ1 (2) = 2, θ2 (2) = 0. All angles are in radians. These boundary conditions when transformed to the space of flat variables give: y1 (0) = 1, y2 (0) = 1.8; y˙ 1 (0) = 0.1, y˙ 2 (0) = −0.065; y1 (2) = 2, y2 (2) = 1.2; y˙ 1 (2) = 0.5, y˙ 2 (2) = −0.3512. The flat output trajectories using these boundary conditions are shown

where γ3

=

γ4

=

(a0 +a1 cos y1 )2 y˙ 2 c0 a1 y˙ 1 2 1 cos y1 ) − (ac00+a a1 sin y1 y˙ 1

(41)

the system dynamics becomes y¨1 y¨2

= v1 = v2 .

(42)

With the choice vi = y¨id + kvi (y˙id − y˙ i ) + kpi (yid − yi ), i = 1, 2 we can ensure that y(t) will asymptotically follow yid (t) under the proper selection of kvi and kpi . An example of an implementation of the feedback controller is shown in Fig. 4. The initial conditions at time t = 0 were chosen as θ0 = 1.96, θ1 = 0.8, θ2 = −0.1 so that the initial starting point is not on the given reference trajectory. The feedback gains were selected as kv1 = 100, kp1 = 10, kv2 = 110, kp2 = 13. As shown in the plots, the trajectories follow closely the desired trajectory. VII. CONCLUSIONS We have studied systematically free-floating planar space robots. Their motion is characterized by nonintegrable rate constraints. These constraints have

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Fig. 4. The feedback y1d , y2d , θ0d , θ1d , θ2d .

control

performance

[3] N. Faiz S. K. Agrawal, R. M. Murray, “Trajectory Planning of Differentially Flat Systems with Dynamics and Inequalities”, AIAA Journal of Guidance, Control, and Dynamics, vol. 24, no. 2, 2001, pp. 219-227. [4] Fliess, M., Levine, J., Martin, P., Rouchon, P., “Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples”, International Journal of Control, vol. 61, no. 6, 1995, pp. 13271361. [5] L.R. Hunt, R. Su, G. Meyer. “Global transformations of nonlinear systems”, IEEE Transactions on Automatic Control vol. 28, 1983, pp. 24-31. [6] P. Martin, P. Rouchon. “Feedback linearization and driftless systems”. Mathematics of Control, Signal and Systems vol. 7, 1994, pp. 235-254.

about

IX. APPENDIX allowed us to write the equations of the motion as an input-affine drift-less control system. In order to use the main advantage of flatness, which is ease in design of algorithms for trajectory generation and control, design conditions have been determined for the system to be flat. Although, in general, it is not easy to determine if a given system is flat, for driftless systems of codimension one the flatness condition is equivalent to controllability condition. Therefore, the parameters of the robot have been chosen so that the system is controllable and is linearizable by prolongations. This implies the existence of a diffeomorphism between the system variables and the flat variables. Using these design conditions, trajectory has been generated between initial and final states, and a feedback control law eliminates perturbations in the initial state. Even though the result has been presented in the context of a three link planar space robot, the authors have shown that the result extends to a general n-link planar robot where the center of mass of the last link is on the preceeding joint axis.

A = −M y0 − (m0 d0 + m1 l0 + m2 l0 ) sin θ0 −(m1 d1 + m2 l1 ) sin(θ0 + θ1 ) − m2 d2 sin(θ0 + θ1 + θ2 ) B = −M x0 − (m0 d0 + m1 l0 + m2 l0 ) cos θ0 −(m1 d1 + m2 l1 ) cos(θ0 + θ1 ) − m2 d2 cos(θ0 + θ1 + θ2 ) C = m2 l02 + m2 l12 + m2 d22 + m1 l02 + m1 d21 + m0 d20 +(m2 l0 + m1 l0 + m0 d0 )(x0 cos θ0 + y0 sin θ0 )+ 2  +(2m2 l0 l1 + m2 l1 d2 + 2m1 l0 d1 ) cos θ1 + Ii i=0

+(m2 l1 + m1 d1 )(x0 cos(θ0 + θ1 ) + 2m2 l1 d2 cos θ2 +y0 sin(θ0 + θ1 )) + 2m2 l0 d2 cos(θ1 + θ2 ) 2  Ii D = m2 l12 + m2 d22 + m1 d21 + i=1

+(m2 l0 l1 + m1 l0 d1 ) cos θ1 +(m2 l1 + m1 d1 )(x0 cos(θ0 + θ1 ) + y0 sin(θ0 + θ1 ))+ +m2 l0 d2 cos(θ1 + θ2 ) + 2m2 l1 d2 cos θ2 +m2 d2 (x0 cos(θ0 + θ1 + θ2 ) + y0 sin(θ0 + θ1 + θ2 ) E = m2 d22 + I2 + m2 d2 (x0 cos(θ0 + θ1 + θ2 ) +y0 sin(θ0 + θ1 + θ2 ) + m2 l0 d2 cos(θ1 + θ2 ) +m2 l1 d2 cos θ2 a0

ACKNOWLEDGMENTS We acknowledge the research support of National Science Foundation Presidential Faculty Fellow Award.

a1 a2 a3 a4 b0

VIII. REFERENCES [1] S.K. Agrawal, A. Fattah “Reactionless Space and Ground Robots: Novel Designs and Concept Studies,” To appear in Mechanism and Machine Theory, 2003. [2] S. K. Agrawal, G. Hirzinger, K. Landzettel, and R. Schwertassek, “A New Laboratory Simulator for Study of Motion of Free-floating Robots Relative to Space Targets”, IEEE Trans. on Robotics and Automation, vol. 123, no. 4, 1996, pp. 627-633.

b1 b2 b3 c0 c2 c3

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= + = = = = = + = = = = = =

m1 m2 (l1 − d1 )2 + m0 (l0 − d0 )2 (m1 + m2 ) + m2 d22 M (I0 + I1 + I2 )M + m0 (m2 l12 + m1 d21 ) 2m0 (l0 − d0 )(m2 l1 + m1 d1 ) + m2 l1 d2 M m1 m2 d2 (l1 − d1 ) + m0 m2 l1 d2 + m2 l1 d2 M m0 m2 d2 (l0 − d0 ) + m2 l0 d2 M m2 d2 −(m1 m2 (l1 − d1 )2 + (I1 + I2 )M + m0 (m2 l12 + m1 d21 ) m2 d22 (m0 + m1 )) −(m0 (l0 − d0 )(m2 l1 + m1 d1 ) − m0 m2 d0 l1 ) −(2m1 m2 d2 (l1 − d1 ) + 2m0 m2 l1 d2 ) −(m0 m2 d2 (l0 − d0 )) −(m2 d22 (m0 + m1 ) + I2 M ) −(m1 m2 d2 (l1 − d1 ) + m0 m2 l1 d2 ) −(m0 m2 d2 (l0 − d0 ))